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The well-posedness of two-dimensional ideal flow. (English) Zbl 0561.76032

This is a review of mathematical results concerning well-posedness of the Euler equation for an ideal fluid in two dimensions. When the vorticity is initially bounded (in Hölder norm), there exists for all time a unique solution which remains as smooth as the initial data. This condition is not satisfied in the context of the Kelvin Helmholtz instability where the initial velocity is discontinuous through a vortex sheet. In the latter case, existence for a short time of the vortex sheet is insured only if the sheet and the vorticity density are initially analytic. Asymptotic analysis [D. W. Moore, Proc. R. Soc. Lond., Ser. A 365, 105-119 (1979; Zbl 0404.76040)] and numerical simulations [D. I. Meiron, G. R. Baker and S. A. Orszag, J. Fluid Mech. 114, 283-298 (1982; Zbl 0476.76031)] suggest that a singularity in the vortex sheet develops in a finite time. Existence of the flow all time is then ensured only in a weak sense. The emphasis is on an elementary presentation of the methods. We have tried to bring out the hard cores of the proofs (mainly a priori estimates), leaving out some technical details.

MSC:

76B47 Vortex flows for incompressible inviscid fluids
35Q99 Partial differential equations of mathematical physics and other areas of application
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